What is the integral of (10x^9)/(x^{10)+1} ?

The integral of (10x^9)/(x^{10)+1} is ln|x^{10}+1|+C

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integral of (10x^9)/(x^{10)+1}

\int \frac{10 x ^{9}}{x ^{10} + 1} d x

ln x^{10} + 1 + C

Solution steps

\int \frac{10 x ^{9}}{x ^{10} + 1} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int \frac{x ^{9}}{x ^{10} + 1} d x

Apply u-substitution

= 10 \cdot \int \frac{1}{10 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10 \cdot \frac{1}{10} ln ∣ u ∣

Substitute back

u = x^{10} + 1

= 10 \cdot \frac{1}{10} ln x^{10} + 1

Simplify

10 \cdot \frac{1}{10} ln x^{10} + 1 : ln x^{10} + 1

= ln x^{10} + 1

Add a constant to the solution

= ln x^{10} + 1 + C

What is the integral of (10x^9)/(x^{10)+1} ?
The integral of (10x^9)/(x^{10)+1} is ln|x^{10}+1|+C